Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am trying to figure out the form of the summation indices of a multi-binomial expansion. Suppose the condensed form is

$$ \begin{align} S &= \prod_{a=1}^{N-1} \prod_{b=a+1}^{N} (1+x_{ab}) \\ &= \prod_{a<b} (1+x_{ab}) \\ &= (1+x_{12})(1+x_{13})(1+x_{14})\cdots(1+x_{23})(1+x_{24})\cdots(1+x_{N-1,N}) \end{align}.$$

This can be expanded as

$$ \begin{align} S &= (1+x_{12}+x_{13}+x_{12}x_{13})(1+x_{14})\cdots(1+x_{N-1,N})\\ &=(1+x_{12}+x_{13}+x_{14}+x_{12}x_{13}+x_{12}x_{14}+x_{13}x_{14}+x_{12}x_{13}x_{14})\cdots(1+x_{N-1,N})\\ &=1+\sum_{a<b} x_{ab} + \sum_{?} x_{ab}x_{cd} + \sum_{?}x_{ab}x_{cd}x_{ef} + \cdots + x_{12}x_{13}x_{14}\cdots{}x_{N-1,N} \end{align}.$$

I am trying to figure out how to express the indices of the sums of higher order (e.g., those containing question marks in the above expression). For instance, the indices in the term $\displaystyle\sum_{?} x_{ab}x_{cd}$ cannot be $\displaystyle\sum_{\substack{a<b\\c<d}} x_{ab}x_{cd}$ because this double counts terms. Nor can it be $\displaystyle\sum_{a<b<c<d} x_{ab}x_{cd}$ because this misses terms. How can I express the indices of this sum and the higher order sums in the above expression?

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.